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Abstract:

We develop and estimate a general equilibrium model that
accounts for key business cycle properties of macroeconomic
aggregates, including labor market variables. In sharp contrast to
leading New Keynesian models, wages are not subject to exogenous
nominal rigidities. Instead we derive wage inertia from our
specification of how firms and workers interact when negotiating
wages. Our model outperforms the standard
Diamond-Mortensen-Pissarides model both statistically and in terms
of the plausibility of the estimated structural parameter values.
Our model also outperforms an estimated sticky wage model.

1. Introduction

Employment and unemployment fluctuate a great deal over the
business cycle. Macroeconomic models have difficulty accounting for
this fact. See for example the classic real business cycle models
of Kydland and Prescott (1982) and Hansen (1985). Models that build
on the search-theoretic framework of Diamond (1982), Mortensen
(1985) and Pissarides (1985) (DMP) also have difficulty accounting
for the volatility of labor markets, see Shimer (2005a). In both
classes of models, the problem is that real wages rise sharply in
business cycle expansions, thereby limiting firms' incentives to
expand employment. The proposed solutions depend on controversial
assumptions, such as high labor supply elasticities or high
replacement ratios.1

Empirical New Keynesian models have been relatively successful
in accounting for the cyclical properties of employment. However,
they do so by assuming that wage-setting is subject to nominal
rigidities and that employment is demand-determined.2 These
assumptions prevent the sharp rise in wages that limits the
employment responses in standard models. Empirical New Keynesian
models have been criticized on at least four grounds. First, these
models do not explain wage inertia, they just assume it. Second,
agents in the model would not choose the wage arrangements that are
imposed upon them by the modeler.3 Third, empirical New Keynesian
models are inconsistent with the fact that many wages are constant
for extended periods of time. In practice, these models assume that
agents who do not reoptimize their wage simply index it to
technology growth and inflation.4 So, these models predict that all
wages are always changing. Fourth, these models cannot be used to
examine some key policy issues such as the effects of an extension
of unemployment benefits.

In this paper we develop and estimate a model that accounts for
the response of key macro aggregates, including labor market
variables like wages, employment, job vacancies and unemployment to
identified monetary policy shocks, neutral technology shocks and
investment-specific technology shocks. In contrast to leading
empirical New Keynesian models, we do not assume that wages are
subject to exogenous nominal rigidities. Instead, we derive wage
inertia as an equilibrium outcome. Like empirical New Keynesian
models, we assume that price setting is subject to exogenous
nominal Calvo-style rigidities. Guided by the micro evidence on
prices, we assume that firms which do not reoptimize their price
must keep it unchanged, i.e. no price indexation.

We take it as given that a successful model must have the
property that wages are relatively insensitive to the aggregate
state of the economy. Our model of the labor market builds on Hall
and Milgrom (2008), henceforth HM.5 In practice, by the time
workers and firms sit down to bargain, they know there is a surplus
to be shared if they can come to terms. So, rather than just going
their separate ways in the wake of a disagreement, workers and
firms continue to negotiate.6 This process introduces a delay
in the time required to make a deal. This delay is costly for both
workers and firms. HM's key insight is that if the associated costs
are relatively insensitive to the aggregate state of the economy,
then negotiated wages will inherit that insensitivity.

This paper investigates whether a dynamic general equilibrium
model which embeds this source of wage inertia can account for key
business cycle properties of labor markets. We show that it does.
In the wake of an expansionary shock, wages rise by a relatively
small amount, so that firms receive a substantial fraction of the
rents associated with employment. Consequently, firms have a strong
incentive to expand their labor force. In addition, the muted
response of wages to aggregate shocks makes firms' marginal costs
relatively acyclical. This acyclicality of marginal costs enables
our model to account for the inertial response of inflation even
with modest exogenous nominal rigidities in prices.

In our benchmark model, we assume that workers and firms bargain
over the current wage rate in each period, taking as given the
outcome of future wage negotiations (period-by-period
bargaining). We also consider an alternative approach in which
firms and workers bargain just once, when they first meet. At that
time, they bargain over the present discounted value of the wages
that prevail throughout the duration of their match (present
discounted value bargaining). In this approach, firms and
workers are indifferent about how wages are paid out over dates and
states of nature, as long as the present discounted value of wage
payments is consistent with the outcome of their negotiation. We
show that the two approaches to bargaining lead to identical
equilibrium allocations, though to possibly different spot wages.
For example, present discounted value bargaining is consistent with
the nominal wage of an individual worker being constant for
extended periods of time and wages of job changers being more
volatile than those of incumbent workers.

We adopt period-by-period bargaining as our benchmark
specification. First, it allows us to incorporate wage data into
our empirical analysis. Second, the present discounted value
bargaining makes strong assumptions about agents' ability to commit
to streams of wage payments. We defer analyzing the time
consistency of these commitments to future work.

We estimate our model using a Bayesian variant of the strategy
in Christiano, Eichenbaum and Evans (2005), henceforth CEE. That
strategy involves minimizing the distance between the dynamic
response to three shocks in the model and the analog objects in the
data. The latter are obtained using an identified vector
autoregression (VAR) for 12 post-war, quarterly U.S. times series
that includes key labor market variables. We contrast the empirical
properties of our model with estimated versions of leading
alternatives. The first alternative is a variant of our model in
which the labor market corresponds closely to the standard DMP
model. The second alternative is a version of the standard New
Keynesian sticky wage model of the labor market proposed in Erceg,
Henderson and Levin (2000), henceforth EHL. The version that we
emphasize does not allow for wage indexation because the resulting
implications are strongly at variance with micro data on nominal
wages of incumbent workers. For comparability we also report
results for a version of the sticky wage model with wage
indexation.

We show that our model outperforms the DMP model in terms of
both model fit and the plausibility of the estimated structural
parameter values. For example, in the estimated DMP model, the
replacement ratio of income for unemployed workers is substantially
higher than the upper bound suggested by existing microeconomic
evidence. Our models also outperforms the DMP model based on the
metrics adopted in the labor market search literature. Authors like
Shimer (2005a) emphasize that the standard deviation of labor
market tightness (vacancies divided by unemployment) is orders of
magnitude higher than the standard deviation of labor productivity.
Our model has no difficulty in accounting for the statistics that
Shimer (2005a) emphasizes.

Finally, we show that our model outperforms the sticky wage New
Keynesian model with no wage indexation in terms of statistical
fit. The statistical fit of the model with indexation is marginally
worse than our model. We conclude that given the limitations of the
sticky wage models, there is simply no need to work with them. The
alternating offer bargaining model has stronger micro foundations,
fits the data better and can be used to analyze a broader set of
labor market variables, e.g. job vacancies and job finding
rates.

Our paper is organized as follows. Section 2 describes the labor
market of our model in isolation. Section 3 integrates the labor
market model into a simple New Keynesian model without capital. We
use this model to discuss the intuition about how our model of the
labor market works in a general equilibrium setting with sticky
prices. Section 4 describes our empirical model. Section 5
describes our econometric methodology. Section 6 presents our
empirical results. Section 7 contains concluding remarks.7

2. The Labor Market

In this section we discuss our model of the labor market. We
assume there is a large number of identical, competitive firms that
produce a homogeneous good using labor. Let
denote the marginal revenue from
hiring an additional worker. Here, we treat
as an exogenous stochastic
process. In the next section, we embed the labor market in a
general equilibrium model and determine the equilibrium process for

At the beginning of period a firm pays a fixed
cost, , to meet a worker with probability one.
We refer to this specification as the hiring cost specification.
Once a worker and a firm meet, they engage in bilateral bargaining.
If bargaining results in agreement (as it always does in
equilibrium) the worker begins production immediately.

We denote the number of workers employed in period by The size of the labor force is
normalized to one. At the end of the period, a fraction of randomly selected employed workers is separated
from their firm. These workers join the ranks of the unemployed and
search for work. So, at the end of the period, there are
workers searching for a job. In
period a fraction, , of
searching workers meet a firm and the complementary fraction
becomes unemployed. With probability a worker
who is employed at time remains with the same firm
in period With probability
this worker
moves to another firm in period Finally, with
probability
this worker
is unemployed in period Our measure of
unemployment in period is
We think of workers that change jobs between
and as job-to-job movements in employment.
There are
workers of
this type. With our specification, the job-to-job transition rate
is substantial and procyclical, consistent with the data (see
Shimer, 2005b). While controversial, the standard assumption that
the job separation rate is acyclical has been defended on empirical
grounds (see Shimer, 2005b). 8 Finally, we think of the time
period as one quarter.

Let denote the expected present
discounted value of the wage payments by a firm to a worker that it
is matched with:

(2.1)

Here denotes the time
wage rate. The discount factor is an
exogenous stochastic process . In our general equilibrium model
(see sections 3 and 4),
we determine the endogenous equilibrium process for . Let denote the value to a firm of
employing a worker in period :

(2.2)

where
denotes the expected
present discounted value of

(2.3)

Because there is free entry into the labor market, firm profits
must be zero. It follows that,

(2.4)

We denote by the value to a worker of being
matched with a firm that pays in period

(2.5)

Here, denotes the value of working
for another firm in period . In equilibrium,
. Finally, in (2.5) is the value of being an
unemployed worker in period It is convenient
to rewrite (2.5) as follows:

(2.6)

where

(2.7)

Note that consists of two components. The
first is the expected present value of the wages received by a
worker from a firm that he is matched with at time .
The second corresponds to the expected present value of the
payments that a worker receives in all dates and states when he is
separated from that firm.

The value of unemployment, , is given
by,

(2.8)

where
denotes the continuation value
of unemployment:

(2.9)

In (2.8), denotes goods
received by an unemployed worker from the government.

The number of employed workers evolves as follows:

(2.10)

Here denotes the hiring rate so that the
number of new hires in period is equal to
The job finding rate is given
by,

(2.11)

The numerator is the number of newly hired workers at the beginning
of time The denominator is the number of workers
who are searching for work at the end of time

2.1 Wage Determination: Alternating Offer
Bargaining

Our baseline specification assumes period-by-period bargaining.
That is, we assume that workers and firms bargain in period
over the period wage rate,
taking as given the outcome of future bargains that will occur as
long as they remain matched. Future wage agreements matter for
current negotiations via their present discounted value,
:

(2.12)

The bargaining problem of all workers is the same, regardless of
how long they have been matched with a firm. This result follows
from our assumptions that hiring costs are sunk at the time of
bargaining and the expected duration of a match is independent of
how long a match has already been in place.

Consistent with Hall and Milgrom (2008), wages are determined
according to the alternating offer bargaining protocol proposed in
Rubinstein (1982) and Binmore, Rubinstein and Wolinsky (1986). Each
time period (a quarter) is subdivided into periods
of equal length, where is even. Firms make a wage
offer at the start of the first subperiod. They also make offers at
the start of every subsequent odd subperiod in the event that all
previous offers have been rejected. Similarly, workers make a wage
offer at the start of all even subperiods in case all previous
offers have been rejected. Because is even, the
last offer is made, on a take-it-or-leave-it basis, by the worker.
In subperiod
the recipient of an offer can
either accept or reject it. If the offer is rejected the recipient
may declare an end to the negotiations or he may plan to make a
counteroffer at the start of the next subperiod. In the latter case
there is a probability, that bargaining breaks
down.

Consider a firm that makes a wage offer,
in subperiod for odd. The firm
sets as low as possible subject to the
condition that the worker does not reject it. Other things equal,
the firm would like to make an offer that worker accepts because a
lack of agreement delays the onset of production. So, it is optimal
for the firm to offer the lowest wage subject to the worker not
rejecting it. The resulting wage offer,
satisfies the following indifference condition on the part of the
worker:

(2.13)

We assume that when an agent is indifferent between accepting and
rejecting an offer, he accepts it. The left hand side of (2.13), denotes the value to
a worker of accepting the wage offer

(2.14)

where
and are
taken as given by a worker-firm bargaining pair. The right hand
side of (2.13) represents the worker's
disagreement payoff, i.e. the value to a worker of rejecting the
wage offer, with the intention of making a counteroffer. The first
term on the right hand side of (2.13) reflects
the possibility that negotiations exogenously break down and the
worker becomes unemployed. The value of becoming unemployed at time
in subperiod is given by

where the term involving reflects our assumption that
the worker receives unemployment benefits in period
in proportion to the number of subperiods spent in unemployment.

The second term on the right hand side of (2.13) reflects the fact that with probability
the worker will receive unemployment
benefits for a period and make a counteroffer
to the firm which he expects to be
accepted. Equation (2.13) represents the
relevant indifference condition assuming that the worker's
disagreement payoff exceeds the value of his outside option,
. In practice one must verify that
this condition holds.

Next, consider the problem of a worker who makes an offer in
subperiod, where and
is even. Other things equal, the worker
would like to make an offer that the firm accepts because a lack of
agreement delays the payment of wages whose value exceeds
unemployment benefits. So, it is optimal for the worker to offer
the highest wage subject to the firm not rejecting it. The
resulting wage offer, satisfies the
following indifference condition on the part of the firm:

(2.15)

The left hand side of (2.15) denotes the value
to a firm of accepting the wage offer

(2.16)

where

(2.17)

The first term on the right hand side of (2.16) is
the value to the firm of accepting the worker's offer and beginning
production immediately. The term
in (2.16) reflects our assumption that one worker produces
goods in each subperiod during which
production occurs. The term
represents the
expected present value of future marginal revenues associated with
a worker, while
represents the expected present value of wage payments to the
worker if the firm accepts the wage offer

The expression on the right side of (2.15) is
the firm's disagreement payoff. If the firm rejects the worker's
offer with the intention of making a counteroffer there is a
probability, , that negotiations break down and
the firm goes to its outside option whose value is zero. With
probability the firm makes a counteroffer,
in the next subperiod. To make a
counteroffer, the firm incurs a cost, . The
second expression in the square bracketed term in (2.15) reflects that the value associated with a
successful firm counteroffer, . Equation
(2.15) represents the relevant indifference
condition governing the worker's wage offer assuming that the
firm's disagreement payoff exceeds the value of its outside option,
i.e., zero. In practice one must verify that this condition
holds.

Finally, consider subperiod in which the
worker makes a take-it-or-leave-it offer. The worker chooses the
highest possible wage subject to the condition that the firm does
not reject it and go to the outside option. Since the latter has
zero value, we can write the firm's indifference condition
as:9

(2.18)

where

(2.19)

We now summarize how to compute the equilibrium wage rate. Note
that and
always appear as a sum in
the equilibrium conditions, (2.13), (2.15) and (2.18),

(2.20)

for
We can solve for given the variables that are exogenous to the
worker-firm pair. Then, (2.13) for can be solved for
and (2.15)
can be solved for
In this way, the equilibrium
conditions can be used to solve uniquely for a set of values

(2.21)

conditional on variables that are exogenous to the worker-firm
bargaining pair. The equilibrium present discounted value of the
wage,
is just
Because (2.13), (2.15) and (2.18) are simple linear equations, they can be solved
analytically for

(2.22)

where

It can be shown that for are
strictly positive. Appendix A contains a detailed derivation of the
previous equation.

Finally, we use (2.1) and (2.12) to compute the period wage rate
conditional on and a given set of beliefs about future wages as
summarized by
. Our analysis indicates
that is uniquely determined conditional
on the variables that are exogenous to the worker-firm pair,
In principle the additive
decomposition of into the current wage
rate, , and future payments summarized by
is not uniquely
determined. We resolve this potential non-uniqueness in the timing
of wage payments by our assumption that wages in each date are the
same time-invariant function of a small set of state variables.

Given laws of motion for
and
a period-by-period bargaining equilibrium is a stochastic process
for the ten variables,

In the following subsection, we exploit the block recursive
structure of this equilibrium, i.e. the fact that the first
eight variables in (2.23) can be computed without
reference to and
.

We conclude this subsection by noting that in the standard DMP
setup, is determined by the following
Nash sharing rule:

(2.24)

where
is the share of the total
surplus given to workers. It is straightforward to show that
(2.22) can be re-written as an Alternating
Offer Bargaining sharing rule:

(2.25)

where
for
So, as in the Nash sharing rule,
depends on and
with weights
determined by the parameters describing the environment of the
model economy. However, there are two constant terms involving
and that are, by
assumption, not a function of the state of the economy, as well as
a separate term in
In section 3 we provide intuition for how the parameters of
agents' environment affect the sensitivity of wages to different
shocks to the economy.

2.2 Implications for Wages

In our baseline model bargaining occurs on a period-by-period
basis. We now consider the present discounted value bargaining
arrangement and show that it leads to the identical quantity
allocation as period-by-period bargaining. Moreover the resulting
equilibrium is consistent with the following empirical
observations: (i) incumbent workers and firms do not renegotiate
wages every period; (ii) the nominal wage of incumbent workers is
often constant for extended periods of time; and (iii) the
volatility of wages paid to new hires is higher than the volatility
of wages paid to incumbent workers.

Under present discounted value bargaining, workers and firms
only bargain once, namely when they first meet. The negotiations
pertain to the wage that the firm will pay to the worker in each
date and state of nature where they remain matched. Since workers
and firms only care about the present discounted value of wages, we
suppose that they begin by bargaining over the scalar,
The structure of bargaining
parallels the period-by-period bargaining framework discussed in
the previous subsection, with obvious modifications.

As before, we divide the quarter in which the firm and worker
first meet into subperiods. At the start of the
first subperiod, the firm makes an offer, which we denote by
. The firm makes the lowest
possible offer subject to the constraint that the offer is not
rejected by the worker. This constraint implies that
satisfies a version of (2.13) in which
in (2.14) is replaced by
for To
evaluate the right side of (2.13) the firm must
know the worker's counteroffer,
in case the worker rejects the
firm's offer. The worker's counteroffer satisfies a version of
(2.15) with
in (2.16) replaced by
for But, to
evaluate the right side of (2.15) with
the worker must know the firm's
counteroffer,
in case the firm rejects
, and so on. It follows that to
make its initial offer
the firm must solve for
for all
If all offers
are rejected, then the worker
makes a final take-it-or-leave-it offer,
This offer has the property
that a version of (2.18) holds, the modification
being that
in (2.19) is replaced with

The solution to this sequence of equations is a unique set,
(2.21), which can be computed by iterating
through the subperiods beginning with and
working backwards. By construction, the offer
is accepted and corresponds to
the equilibrium present discounted value of the wage,
The equations
whose solution yields are exactly the
same as the equation used to solve the period-by-period bargaining
problem. So, the solution has the same characterization, (2.22).

An agreement between a worker and firm constitutes a sequence of
wage rates indexed by each of the dates and states of nature over
the duration of their match, subject to the constraint that the
sequence is consistent with the agreed-upon value of
We denote by
the sequence of date and
state contingent wage rates that the firm and the worker agree on.
We think of a given specification of
as a particular wage payment
scheme. Given laws of motion for
and a
present discounted value bargaining equilibrium is a set of nine
stochastic processes,

(2.26)

The first eight of these stochastic processes satisfy the same
eight conditions described after (2.23) and the
wage payment scheme
must satisfy the restriction
that the expected present value of its elements is equal to

Suppose that we have a set of the ten objects in (2.23) that satisfy the relevant equilibrium conditions.
The first eight of these objects satisfy the same conditions
required of the first eight objects in (2.26). Also let
consist of the sequence of
wage rates in the period-by-period bargaining equilibrium. The
expected present value of that sequence of wage rates is
We conclude:

An implication of this proposition is that the equilibrium in
our baseline model is consistent with empirical observation (i),
namely incumbent workers and firms do not renegotiate wages every
period. This consistency follows because we can interpret the
period-by-period bargaining equilibrium as a particular present
discounted value bargaining equilibrium in which the wage rates in
each date and state of nature correspond to the relevant wage rates
in our period-by-period bargaining equilibrium.

The equilibrium in our baseline model is also in principle
consistent with empirical observations (ii) and (iii), namely that
the nominal wage of incumbent workers often does not change for
long periods of time and the wages of newly hired workers are more
volatile than those of incumbent workers. The reason is that under
present discounted value bargaining, the following payment scheme
is an equilibrium: a worker that bargains in period
receives a fixed nominal wage payment, for
each date and state in which the match continues. This equilibrium
is clearly consistent with observations (ii) and (iii).

A potential difficulty with the way that we account for
observations (ii) and (iii) is that the equilibrium may not be time
consistent. By time consistency of a present value bargaining
equilibrium we mean that if present value bargaining was re-started
at some time in the future, the firm and worker would again agree
on the same state and date contingent wage rates they agreed on
when they first met. Clearly not all present discounted value
bargaining equilibria are time consistent. For example, suppose
that
involves the firm paying
to the worker in period
and zero thereafter. If bargaining were
re-opened at a later date, the worker would not accept a zero wage
payment. So, in general, present discounted value bargaining
requires that we make strong assumptions about agents' ability to
commit.

In this section we incorporate the labor market model of the
previous section into the benchmark New Keynesian macroeconomic
model using a structure that is very similar to Ravenna and Walsh
(2008). We use this framework to explore the intuition for how the
alternating offer bargaining model of the labor market helps to
account for the cyclical behavior of key macroeconomic
variables.

3.1. Simple Framework

As in Andolfatto (1995) and Merz (1996), we assume that each
household has a unit measure of workers. Because workers experience
no disutility from working, they supply their labor inelastically.
An employed worker brings home the real wage, An unemployed worker receives goods
in government-provided unemployment compensation. Unemployment
benefits are financed by lump-sum taxes paid by the household.
Workers maximize their expected income, subject to the labor market
arrangements described in the previous section. By the law of large
numbers, this strategy maximizes the total income of the household.
Workers maximize expected income in exchange for perfect
consumption insurance from the household. All workers have the same
concave preferences over consumption. So, the optimal insurance
arrangement involves allocating the same level of consumption,
to each worker.

The household maximizes:

subject to the budget constraint:

Here
denotes the fraction of
household members who are employed. In addition, denotes lump-sum taxes net of lump-sum transfers and
profits. Also denotes purchases of bonds in
period and denotes the
gross nominal interest rate on bonds purchased in the previous
period. Finally, the variables and
denote the nominal wage rate and price
of the final good.

A final homogeneous good, is produced
by competitive and identical firms using the following
technology:

(3.1)

where
The representative firm
chooses specialized inputs, to maximize
profits:

subject to the production function (27). The
firm's first order condition for the input
is:

(3.2)

As in Ravenna and Walsh (2008), the input
good is produced by a monopolist retailer, with production
function,

where is the quantity of the intermediate
good purchased by the producer. This
intermediate good is purchased in competitive markets at the
after-tax price
from a
wholesaler. Here, represents a subsidy
(financed by a lump-sum tax on households) which has the effect of
eliminating the monopoly distortion in the steady state. That is,
where denotes the steady state markup. In the retailer
production function, denotes a technology
shock that has the law of motion:

where
is the shock to technology and

The monopoly producer of sets
subject to Calvo sticky price
frictions. In particular,

(3.3)

Here, denotes the optimal price set
by the fraction of producers who have the
opportunity to reoptimize. Note that we do not allow for price
indexation. So, the model is consistent with the observation that
many prices remain unchanged for extended periods of time (see
Eichenbaum, Jaimovich and Rebelo, 2011, and Klenow and Malin,
2011).

Let,

(3.4)

where
so that
denotes the retail firm's real
marginal cost. Also, let

The wholesalers who produce correspond to
the perfectly competitive firms modeled in the previous section.
Recall that they produce using labor only and
that labor has a fixed marginal productivity of unity. The total
supply of the intermediate good is given by
which equals the total quantity of labor used by the wholesalers.
So, clearing in the market for intermediate goods requires

(3.5)

We adopt the following monetary policy rule:

(3.6)

where
denotes the gross
inflation rate and
is a monetary policy
shock. In addition, a time series variable without a time subscript
refers to its value in nonstochastic steady state.

3.2. Integrating the Labor Market into the Simple Framework

There are four points of contact between the model in this
section and the one in the previous section. The first point of
contact is the labor market in the wholesale sector where the real
wage is determined as in section 2. The second
point of contact is via
in (3.4),
which corresponds to the real price that appears in the previous
section (see (2.3)). The third point of contact
occurs via the asset pricing kernel, which
is given by:

(3.7)

The fourth point of contact is the resource constraint which
specifies how the homogeneous good, is
allocated among its possible uses:

(3.8)

where

(3.9)

Here,
denotes the cost of
generating new hires in period The expression on
the right side of (3.9) is the production
function for the final good. The absence of price distortions in
this expression reflects Yun's (1996) result that these distortions
can be ignored in (3.9) when linearizing around
a nonstochastic steady state without price distortions.

From the perspective of the model in this section, the prices in
the previous section correspond to real prices. So, corresponds to the real wage rate, where conversion to
real is accomplished using That is,
workers and firms bargain over real wages according to the
alternating wage offer arrangement described in section 2. See the technical appendix for a list of the
equilibrium equations.

3.3. Quantitative Results in the Simple
Model

This subsection displays the dynamic response of our simple
model to monetary policy and technology shocks. In addition, we
discuss the sensitivity of these responses to the wage bargaining
parameters,
and The
first subsection below reports a set of baseline parameter values
for the model. The second subsection presents and discusses impulse
responses.

3.3.1. Baseline Parameterization

Table 1 lists the baseline parameter values. The values for
parameters that are common to the simple macro model and the
medium-sized DSGE model are equal to the prior means that we use
when we estimate the parameters of the latter model. We set the
parameters of the monetary policy rule, (3.6), equal to 1.7, 0.1 and 0.7, respectively. We set the discount factor to 1.03-0.25
so that the implied steady
state real interest rate is the same as in the medium-sized DSGE
model. We assume the steady state gross markup, to be 1.2 and set the degree of
price stickiness, to our prior mean of
0.66. In addition, we set to 0.005, which is the value used by
HM. The parameter is equal to 60
which roughly corresponds to the number of business days in a
quarter. We assume which implies a
match survival rate that is consistent with both HM and Shimer
(2012a).10

We calibrate three model parameters, and to hit three steady state
targets. In particular, we require (i) a steady state unemployment
rate, of 5.5%; (ii) a steady
state value of 1 percent for the ratio of hiring costs to gross
output, i.e.,
and (iii) a steady state
value of 0.4 for the replacement ratio, . The resulting values for and are reported in Table 2 which
also summarizes other steady state properties of the model. The
calibrated value for implies that the firm
must pay 0.61 of a day's worth of the revenue to
generate a counteroffer.

We assume that the AR(1) parameter for the law of motion for
technology, , is equal to 0.95. Finally, for simplicity, we assume that the gross
steady state inflation rate, is equal to
unity.

3.3.2. Impulse Responses

Figures 1 and 2 display the dynamic responses of the model
economy to monetary policy and technology shocks, respectively. We
report results for the baseline parameterization. In addition, we
display results for four other parameterizations, each of which
changes the value of one parameter relative to the baseline case.
In particular we raise from 0.005 in the baseline parameterization to 0.0075. Also, we lower from
0.01 to 0.009, we decrease
from 0.396 to 0.376 and we lower from 60
to 50.

Figure 1 displays the dynamic response of the model economy to a
negative 25 annualized basis point monetary policy shock,
. In the baseline model,
real wages respond by a relatively small amount with the peak rise
equal to 0.03 percent. Inflation also responds by a
relatively small amount, with a peak rise of 0.06 percent (on an annual basis). At the same time, there is
a substantial increase in consumption, which initially jumps by
about 0.13 percent. Finally, the unemployment rate
drops by 0.14 percentage points in the impact period
of the shock.

The basic intuition for how a monetary policy shock affects the
economy in our model is as follows. As in standard New Keynesian
sticky price models, an expansionary monetary policy shock drives
the real interest rate down, inducing an increase in the demand for
final goods. This rise induces an increase in the demand for the
output of sticky price retailers. Since they must satisfy demand,
the retailers purchase more of the wholesale good. Therefore, the
relative price of the wholesale good increases and the marginal
revenue product (
associated with a worker
rises. Other things equal, this motivates wholesalers to hire more
workers and increases probability that an unemployed worker finds a
job. The latter effect induces a rise in workers' disagreement
payoffs. The resulting increase in workers' bargaining power
generates a rise in the real wage. Given our assumptions about
parameter values, alternating offer bargaining mutes the increase
in real wages, thus allowing for a large rise in employment, a
substantial decline in unemployment, and a small rise in inflation.
If the rise in the real wage was large, the incentive of employers
to hire more workers would be weaker and a monetary policy shock
would have less of an expansionary effect.

To provide intuition for the quantitative role of alternating
offer bargaining, Figure 1 displays the economy's response to a
monetary policy shock for different values of
and . To
understand how the model economy responds to shocks, it is useful
to use the value of unemployment, as an
indicator of general economic conditions. Shocks that expand
economic activity tend to simultaneously raise . In what follows, we provide intuition about how the
parameters governing the alternating offer sharing rule,
and influence the responsiveness of the
wage to To do so, we
consider a bargaining session between a single worker and a single
firm. We consider the response of the wage negotiated by this
firm-worker pair to a rise in experienced
idiosyncratically by that pair. For convenience we assume the
experiment occurs when the economy is in nonstochastic steady
state. By this we mean a situation in which all aggregate shocks
are fixed at their unconditional means, aggregate variables are
constant and there is ongoing idiosyncratic uncertainty at the
worker-firm level.

Let denote the particular worker-firm pair
under consideration. Let denote the value of
unemployment to the worker in the
worker-firm pair. The variable, denotes the
wage negotiated by the worker-firm pair. The
object of interest is
the elasticity of with respect to where

(3.10)

In what follows, we assume that firm and worker disagreement
payoffs exceed the value of their outside options. In (3.10), and denote
the economy-wide average value of the wage rate and of the value of
unemployment, respectively, in nonstochastic steady state.

Consider the impact of reducing . A
decrease in raises the disagreement payoff of
the firm, putting the worker in a weaker bargaining position. So,
other things equal, a fall in leads to a
decrease in . As it turns out, this decrease is
the same, regardless of the value of so that
is independent of It follows that affects
entirely through its effect on
The zero profit condition for firms
implies that the steady state value of the real wage is independent
of the bargaining parameters. So, affects
only through its impact on
. A decrease in places
downward pressure on all worker-firm pair wages and
therefore on Since does not
respond to the value of
must rise to neutralize the downward pressure on .
But the rise in leads to a rise in
Consistent with this intuition,
Figure 1 shows that a decrease in does
not affect the steady state real wages but does increases the
steady value of employment and consumption. At the same time it
reduces steady state unemployment relative to the baseline case.
Also consistent with the previous intuition, the fall in
increases the response of real wages
and inflation to a monetary policy shock while it leads to a
smaller change in employment and consumption.

The intuition for the effect of a decline in
is very similar to the intuition for the effect of a fall in
A decline in leaves
the steady real wage unaffected but leads to a fall in steady state
unemployment. Other things equal, a fall in puts
the worker in a worse bargaining position and leads to a fall in
. As it turns out, the decrease in
is the same for all values of
, so that is
independent of Consequently,
affects only through its impact on
. A decrease in places
downward pressure on all worker-firm pair wages and
therefore on Since does not
respond to a change in , the value of must rise to neutralize the downward pressure on
A rise in increases
the worker's disagreement payoff and his bargaining power, thereby
exerting countervailing upward pressure on . This
reasoning underlies the intuition for why a decrease in leads to a rise in and
Consistent with this intuition,
Figure 1 shows that a decrease in does not
affect the steady state real wage, increases the steady value of
employment and consumption, and reduces steady state unemployment.
Also, as the previous intuition suggests, the fall in increases the response of real wages and inflation to a
monetary policy shock while it decreases the responses of
employment and consumption to that shock.

Consider next the impact of increasing .
In terms of the steady state, consumption rises, unemployment
falls, while inflation and the real wage are unaffected. Figure 1
shows that the dynamic responses of the real wage and inflation to
an expansionary monetary policy shock are stronger than in the
baseline case. At the same time, consumption and unemployment
respond by less than in the baseline case. To understand the basic
intuition for the dynamic responses, note that shocks which
increase the value of workers' outside option, as summarized by
tend to increase the real wage rate,
and dampen the effects of an expansionary shock. In the extreme
case of there is no chance that workers
and firms are thrown to their outside options during negotiations.
So, the value of unemployment, simply does
not directly enter into the indifference conditions (2.13), governing workers' and firms' offers. As a
result, the real wage should not depend much on cyclical shocks
that affect . By continuity, a rise in
increases the importance of
in worker's disagreement payoff which
make the real wage more sensitive to shocks. The stronger response
of the real wage reduces the incentive of firms to hire workers,
thereby limiting the expansionary effects of the shock. Finally,
the larger rise in the real wage places upward pressure on the
marginal costs of retailers, leading to higher inflation than in
the baseline parameterization.

Now consider the impact of a lower value of
Figure 1 indicates a fall in leads to stronger
dynamic responses of the real wage and inflation to an expansionary
monetary policy shock. To understand this result consider the
extreme case where is very large. Equations
(2.16) and (2.18) imply

Then (2.2), (2.4), (2.12) and (2.17) imply that

(3.11)

Suppose that and therefore the interest
rate is constant. Then as goes to
infinity, would become constant
and does not depend on cyclical shocks to the economy. In general,
we would expect that this cyclical insensitivity is inherited by
a
conjecture that is consistent with our numerical results. By
continuity, we expect the real wage to be more sensitive to shocks
when is smaller. This intuition is consistent
with the results reported in Figure 1.

Finally, note that in subperiod , the worker
makes a take-it-or-leave-it offer. The cost to the firm of
rejecting the offer is zero, which is obviously not a function of
the state of the economy. While firms and workers never actually
negotiate in the last subperiod of a quarter, that
out-of-equilibrium possibility still affects the actual real wage
rate. Our intuition suggests that the non state-contingent value to
the firm of rejecting the final take-it-or-leave-it offer made by a
worker, should make the actual real wage less sensitive to a shock.
To pursue this intuition we solved a version of model in which the
firm makes a take-or-leave-it offer in subperiod .
The cost to the worker of rejecting such an offer does, to some
extent, depend on the state of the economy. Consistent with our
intuition, in this version of the model, the real wage is more
sensitive to a policy shock than the baseline model. That said, the
other features of the alternating offer bargaining offer model
still act to mute the response of real wages to the shock and allow
for a substantial expansion in aggregate economic activity.

Figure 2 displays the dynamic responses of our baseline model
and the four alternatives to a 0.1 percent
innovation in technology. In the baseline model, real wages rise
but by a relatively modest amount. Inflation also falls by a modest
amount, with a peak decline of about 0.1 percent
(on an annual basis). Notice that unemployment falls by a
substantial amount with a peak decline of about 0.07 percent. The effect of either lowering and or raising is to make inflation more responsive to the technology
shock while the decline in unemployment is muted relative to the
baseline parameterization.

In sum, in this section we have shown that our labor market
model can potentially account for the cyclical properties of key
labor market variables. In the next section we analyze whether it
actually provides an empirically convincing account of those
properties. To that end we embed it in a medium-sized DSGE model
which we estimate and evaluate.

4. An Estimated Medium-Sized DSGE Model

In this section, we describe a medium-sized DSGE model similar
to one in CEE, modified to include our labor market framework. The
first subsection describes the problems faced by households and
goods producing firms. We discuss the labor market in second
subsection. The third subsection specifies the law of motion of the
three shocks to agents' environment. These include a monetary
policy shock, a neutral technology shock and an investment-specific
technology shock. The last subsection briefly presents a version of
the model embodying the standard DMP specification of the labor
market, i.e. wages are determined by a Nash sharing rule and firms
face vacancy posting costs. In addition, we also present a version
of the model with sticky wages as proposed in EHL. These
alternative models represent important benchmarks for
comparison.

4.1. Households and Goods
Production

The basic structure of the representative household's problem is
the same as in section 3.1. Here, we allow
for habit persistence in preferences, time-varying unemployment
benefits, and the accumulation of physical capital, .

The preferences of the representative household are given by:

The parameter controls the degree of habit formation
in household preferences. We assume
The household's budget
constraint is:

(4.1)

As above, denotes lump-sum taxes net of
transfers and firm profits and denotes the
unemployment compensation of an unemployed worker. In contrast to
(2.8), is exogenously
time-varying to ensure balanced growth. In (4.1), denotes
beginning-of-period purchases of a nominal bond
which pays rate of return, at the start
of period and
denotes the nominal rental rate of capital services. The variable
denotes the utilization rate of
capital. As in CEE, we assume that the household sells capital
services in a perfectly competitive market, so that
represents the
household's earnings from supplying capital services.The increasing
convex function
denotes the cost, in units of
investment goods, of setting the utilization rate to
The variable denotes the nominal price of an investment good and
denotes household purchases of
investment goods.

The household owns the stock of capital which evolves according
to,

The function is an increasing and convex
function capturing adjustment costs in investment. We assume that
and its first derivative are both
zero along a steady state growth path.

As in our simple macroeconomic model, we assume that a final
good is produced by a perfectly competitive representative firm
using the technology, (3.1). The final good
producer buys the specialized input,
from a retailer who uses the
following technology:

(4.2)

The retailer is a monopolist in the product market and is
competitive in the factor markets. Here
denotes the total amount of capital services purchased by firm
. Also, represents
an exogenous fixed cost of production which grows in a way that
ensures balanced growth. The fixed cost is calibrated so that
profits are zero along the balanced growth path. In (4.2), is a technology shock
whose properties are discussed below. Finally, is the quantity of an intermediate good purchased by
the retailer. This good is purchased in
competitive markets at the price from a
wholesaler, whose problem is discussed in the next
subsection. Analogous to CEE, we assume that to produce in period
the retailer must borrow
at the start of the period
at the interest rate The retailer repays
the loan at the end of period when he receives
his sales revenues. The retailer sets its
price, subject to its demand curve, (3.2), and the Calvo sticky price friction
(3.3). Recall that we do not allow for
automatic indexation of prices to either steady state or lagged
inflation.

4.2. Wholesalers and the Labor Market

The structure of the labor market is the same as in section 2. Each wholesaler employs a measure of workers.
Let denote the representative
wholesaler's labor force at the end of A
fraction of these workers separates
exogenously. So, the wholesaler has a labor force of
at the start of period
At the beginning of period the wholesaler selects its hiring rate, which determines the number of new workers that it
meets at time . As in the small macro model, we
assume that the wholesaler's cost of hiring is a linear function of
the hiring rate and is denominated in units of the final
consumption good. The value of a worker to the wholesaler,
is also denominated in units of the
final consumption good. To ensure balanced growth we replace
by an exogenous stochastic process
that is uncorrelated with the state of the economy. We denote the
time value of this process by

To hire
workers, the wholesaler must
post
vacancies. Here
denotes the aggregate vacancy filling
rate which firms take as given and is further described below. We
assume that posting vacancies is costless.

The job finding rate is given by (2.11) where
and denote the
economy-wide value of the corresponding wholesaler-specific
variables. Individual workers view and
as being exogenous and beyond their
control. The values of employment and unemployment, and are denoted in units of
the final good. In order to ensure balanced growth, we replace
by an exogenous stochastic process that is
uncorrelated with the state of the economy. We denote the time
value of this process by .

After setting the firm has access to
workers (see equation (2.10)). Each of these workers engages in bilateral
bargaining with a representative of the firm, taking the outcome of
all other negotiations as given. The equilibrium wage rate,
i.e.
is the outcome of the
alternating offer bargaining process described in section 2. To ensure balanced growth we replace by an exogenous stochastic process that is
uncorrelated with the state of the economy. We denote the time
value of this process by
As before, we verify numerically
that all bargaining sessions conclude successfully with the firm
and worker agreeing to an employment contract. Thus, in equilibrium
the representative wholesaler employs all
workers with which it has met, at wage rate .
Production begins immediately after wage negotiations are concluded
and the wholesaler sells the intermediate good at the real price,
.

To summarize, the labor market equilibrium conditions coincide
with the ones derived in section 2 except that
and are replaced by
and
.

4.3. Market Clearing, Monetary Policy and
Functional Forms

The total amount of intermediate goods purchased by retailers
from wholesalers is:

Recall that the output of intermediate goods produced by
wholesalers is equal to the number of workers they employ. So, the
supply of intermediate goods is As in the
simple model, market clearing for intermediate goods requires
The capital services market
clearing condition is:

Market clearing for final goods requires:

(4.3)

The right hand side of the previous expression denotes the quantity
of final goods. The left hand side represents the various ways that
final goods are used. Homogeneous output,
can be converted one-for-one into either consumption goods, goods
used to hire workers, or government purchases, . In addition, some of is
absorbed by capital utilization costs. Finally, can be used to produce investment goods using a linear
technology in which one unit of the final good is transformed into
units of
Perfect competition in the production of investment goods implies,

The asset pricing kernel, is
constructed using the marginal utility of consumption, which we
denote by

Then,

We adopt the following specification of monetary policy:

Here, denotes the monetary authority's target
inflation rate. The steady state inflation rate in our model is
equal to The shock,
is a unit variance, zero
mean disturbance to monetary policy. Also, and
denote the steady values of
and
The variable,
denotes Gross Domestic
Product (GDP):

We assume that grows exogenously in a
way that is consistent with balanced growth. Working with the data
from Fernald (2012) we find that the growth rate of total factor
productivity is well described by an
process. Accordingly, we assume that
is
We also assume that
follows an AR(1) process. The parameters that control the standard
deviations of both processes are denoted by
The
autocorrelation of
is denoted by

Recall that our model exhibits growth stemming from neutral and
investment-specific technological progress. The variables
,
and
converge to
constants in nonstochastic steady state, where

is a weighted average of the sources of technological progress. If
objects like the fixed cost of production, the cost of hiring, the
cost to a firm of preparing a counteroffer, government purchases,
and unemployment transfer payments were constant, they would become
irrelevant over time. To avoid this implication, it is standard in
the literature to suppose that such objects grow at the same rate
as output, which in our case is given by
An unfortunate implication of this assumption is that technology
shocks of both types immediately affect the vector of objects
It seems hard to justify such an assumption. To avoid this
problem, we proceed as in Christiano, Trabandt and Walentin (2012)
and Schmitt-Grohé and Uribe (2012) who assume that
government purchases, are a distributed lag
of unit root technology shocks, i.e. is
cointegrated with but has a smoother
stochastic trend. In particular, we assume that

where
is
with
and
denotes a distributed lag of
past values of defined by,

(4.4)

Here
are parameters to be
estimated. Note that
grows at the same rate as
in the long-run
When
is very close to zero,
is virtually unresponsive in
the short-run to an innovation in either of the two technology
shocks, a feature that we find very attractive on a priori grounds.
In practice, we constrain the first four diagonal elements of
to be the same. For reasons
discussed below, we found that it is useful to allow
to take on a separate value.

We assume that the cost of adjusting investment takes the form:

Here, and
denote the unconditional growth
rates of and . The
value of
in nonstochastic steady state
is
In addition,
represents a model
parameter that coincides with the second derivative of
, evaluated in steady
state It is straightforward to verify that

We assume that the cost associated with setting capacity
utilization is given by,

where
and
are positive scalars. We
normalize the steady state value of to
one. This pins down the value of
given an estimate of
.

Finally, we discuss how vacancies are determined. We posit a
standard matching function:

(4.5)

where
denotes the total number of
vacancies and denotes the vacancy rate. Given
and we use
(4.5) to solve for
Recall that we defined the total number of vacancies by
. We can solve for the
aggregate vacancy filling rate using

(4.6)

The equilibrium of our model has a particular recursive structure.
We can first solve all model variables, apart from and These two variables can then
be solved for using (4.5) and (4.6).

4.4 Alternative Labor Market Models

In this subsection we consider alternative labor market models
that we include in our DSGE framework. First, we describe our
version of the DMP model, which is characterized by search costs
and a Nash sharing rule. Second, we describe the sticky nominal
wage model of EHL.

4.4.1. The DMP Model

In this subsection, we describe the version of the medium-sized
DSGE model which we refer to as the 'Nash Sharing, Search'
specification. We replace our alternating offer sharing rule,
(2.22) by the Nash sharing rule
(2.24) which we repeat for convenience,

We incorporate DMP-style search costs into our DSGE model as
follows. We assume that vacancies are costly and that posting
vacancies is the only action the firm takes to meet a worker. With
probability a vacancy results in a meeting
with a worker. The aggregate rate at which workers are hired,
depends on the aggregate vacancy
rate, according to (4.6).

The cost of setting the vacancy rate to is
given by:

(4.7)

The probability is determined by the matching
function, (4.5).

Compared to our baseline hiring cost setup, three changes are
required to incorporate the search cost specification into the
medium-sized DSGE model.

First, the free entry/zero profit condition, (2.4) is replaced by:

(4.8)

Free entry in the search cost specification implies that the
marginal cost of posting a vacancy is equal to the expected return.
Second, we add and to
the list of variables that must be simultaneously solved for using
(4.5) and (2.4). The third
change involves replacing the hiring cost term in (4.3) with the vacancy cost term (4.7) in the resource constraint. Doing so we
obtain:

(4.9)

We conclude by discussing an important feature of the search
cost specification. Define labor market tightness as:

(4.10)

Relations (4.5) and (4.6)
imply that takes the following
form,

It follows that the probability of filling a vacancy is decreasing
in labor market tightness.

4.4.2. The Sticky Wage Model

We now describe a modification of the medium-sized DSGE model
which incorporate the sticky nominal wage framework of EHL. We
replace the wholesale production sector with the following
environment. The final homogeneous good, is
produced by competitive and identical firms using technology
(3.1). The specialized inputs used in the
production of are produced by retailers using
capital services and a homogeneous labor input. The final good
producer buys the specialized input,
from a retailer who produces the
input using technology (4.2). Capital
services are purchased in competitive rental markets. In (4.2), refers to the
quantity of a homogeneous labor input that firm
purchases from 'labor contractors'. These contractors produce the
homogeneous labor input by combining a range of differentiated
labor inputs, using the following
technology:

(4.11)

Labor contractors are perfectly competitive and take the wage rate,
of as given. They
also take the wage rate, of the labor type as given. Profit maximization on the part
of contractors yields to the labor demand curve:

(4.12)

Substituting (4.11) into (4.12) and rearranging, we obtain:

(4.13)

Specialized labor inputs are supplied by a large number of
identical households. The representative household has many members
corresponding to each type of labor and
provides complete insurance to all of its members in return for
their wage income. The household's budget constraint is given by
(4.1) except that is equal
to zero. This constraint reflects our assumption that the household
owns the capital stock, sets the utilization rate and makes
investment decisions.

It is optimal for the household to assign an equal amount of
consumption to each of its members. The household's utility
function is given by:

(4.14)

Here, is a positive constant and denotes hours worked by the
member of the household. The wage rate of the type of labor, is
determined outside the representative household by a monopoly union
that represents all -type workers across all
households.

The monopoly union faces Calvo-type nominal rigidities when
setting the wage. With probability the
union can optimize the wage and with
probability it cannot. There is no wage
indexation so that in the latter case, the nominal wage rate is
given by:

(4.15)

The union maximizes the welfare of its members. For a more
detailed exposition of the model and its solution, see CEE.

5. Econometric Methodology

We estimate our model using a Bayesian variant of the strategy
in CEE that minimizes the distance between the dynamic response to
three shocks in the model and the analog objects in the data. The
latter are obtained using an identified VAR for post-war quarterly
U.S. times series that include key labor market variables. The
particular Bayesian strategy that we use is the one developed in
Christiano, Trabandt and Walentin (2011), henceforth CTW.

To facilitate comparisons, our analysis is based on the same VAR
as used in CTW who estimate a 14 variable VAR using quarterly data
that are seasonally adjusted and cover the period 1951Q1 to 2008Q4.
As in CTW, we identify the dynamic responses to a monetary policy
shock by assuming that the monetary authority sees the
contemporaneous values of all the variables in the VAR and a
monetary policy shock affects only the Federal Funds Rate
contemporaneously. As in Altig, Christiano, Eichenbaum and Linde
(2011), Fisher (2006) and CTW, we make two assumptions to identify
the dynamic responses to the technology shocks: (i) the only shocks
that affect labor productivity in the long-run are the innovations
to the neutral technology shock, and the
innovation to the investment-specific technology shock, and (ii) the only shock that affects the price of
investment relative to consumption in the long-run is the
innovation to . These identification
assumptions are satisfied in our model. Standard lag-length
selection criteria lead CTW to work with a VAR with 2 lags.11

There is an ongoing debate over whether or not there is a break
in the sample period that we use. Implicitly, our analysis sides
with those authors who argue that the evidence of parameter breaks
in the middle of our sample period is not strong. See for example
Sims and Zha (2006) and Christiano, Eichenbaum and Evans
(1999).

Given an estimate of the VAR we can compute the implied impulse
response functions to the three structural shocks. We stack the
contemporaneous and 14 lagged values of each of these impulse
response functions for 12 of the VAR variables in a vector,
We do not include the job
separation rate and the size of the labor force because our model
assumes those variables are constant. We include these variables in
the VAR to ensure the VAR results are not driven by an omitted
variable bias.

The logic underlying our model estimation procedure is as
follows. Suppose that our structural model is true. Denote the true
values of the model parameters by
Let
denote the
model-implied mapping from a set of values for the model parameters
to the analog impulse responses in
Thus,
denotes the
true value of the impulse responses whose estimates appear in
According to standard classical
asymptotic sampling theory, when the number of observations,
is large, we have

Here, denotes the true values of the
parameters of the shocks in the model that we do not formally
include in the analysis. Because we solve the model using a
log-linearization procedure,
is not a
function of
However, the sampling
distribution of
is a function of
We find it convenient to express
the asymptotic distribution of
in the following form:

(5.2)

where

For simplicity our notation does not make the dependence of
on
and
explicit. We use a consistent estimator of
Motivated by small sample considerations, that estimator has only
diagonal elements (see CTW). The elements in
are graphed in Figures
3-5 (see the solid lines). The gray areas
are centered, 95 percent probability intervals
computed using our estimate of .

In our analysis, we treat
as the observed data. We specify
priors for and then compute the posterior
distribution for given
using Bayes' rule. This
computation requires the likelihood of
given Our
asymptotically valid approximation of this likelihood is motivated
by (5.2):

(5.3)

The value of that maximizes the above function
represents an approximate maximum likelihood estimator of
It is approximate for three reasons:
(i) the central limit theorem underlying (5.2) only holds exactly as
(ii) our proxy for
is guaranteed to be correct only for
and (iii)
is calculated
using a linear approximation.

Treating the function, as the likelihood
of
it follows that the Bayesian
posterior of conditional on
and is:

(5.4)

Here,
denotes the priors on
and
denotes
the marginal density of

The mode of the posterior distribution of
can be computed by maximizing the value of the numerator in
(5.4), since the denominator is not a
function of The marginal density of
is required for an overall
measure of the fit of our model. To compute the marginal
likelihood, we use the standard Laplace approximation. In our
analysis, we also find it convenient to compute the marginal
likelihood based on a subset of the elements in
(see Appendix B for details).

6. Results

In this section we present the empirical results for our model,
the 'Alternating Offer, Hiring' model. In addition, we report
results for a version of our model with the search cost
specification, the 'Alternating Offer, Search' model and the Nash
sharing model with search or hiring costs, the 'Nash Sharing,
Search' and 'Nash Sharing, Hiring' models. The 'Nash Sharing,
Search' model is our version of the DMP model. Finally, we report
results for the 'Sticky Wage' model.

In the first three subsections we discuss results for the
different models. In the final subsection we assess the models'
ability to account for the statistics that Shimer (2005a) uses to
evaluate the standard DMP model.

We set the values for a subset of the model parameters a priori.
These values are reported in Panel A of Table 3. We also set the
steady state values of five model variables, listed in Panel B of
Table 3. We specify so that the steady
state annual real rate of interest is three percent. The
depreciation rate on capital,
is set to imply an annual
depreciation rate of 10 percent. The values of
and
are determined by the sample
average of real per capita GDP and real investment growth in our
sample. We assume the monetary authority's inflation target is
2.5 percent and that profits of intermediate
good producers are zero in steady state. We set the rate at which
vacancies create job-worker meetings, to
0.7, as in den Haan, Ramey and Watson (2000)
and Ravenna and Walsh (2008). We set the steady state unemployment
rate to the average unemployment rate in our sample, implying a
steady state value of equal to 0.055. As in the simple macro model, we set
equal to 60 and equal to
0.9. Finally, we assume that the steady
state value of the ratio of government consumption to gross output
is 0.20.

All remaining model parameters are estimated subject to the
restrictions summarized in Table 3. Table 4 presents prior and
posterior distributions for all of the estimated objects in the
models.

6.1. The Estimated 'Alternating Offer,
Hiring' Model

A number of features of the posterior mode of the estimated
parameters in the 'Alternating Offers, Hiring' model are worth
noting. First, the posterior mode of implies a
moderate degree of price stickiness, with prices changing on
average once every 2.4 quarters. This value lies
within the range reported in the literature. For example, according
to Nakamura and Steinsson (2012), the recent micro-data based
literature finds that the price of the median product changes
roughly every 1.5 quarters when sales are included,
and every 3 quarters when sales are excluded.
Second, the posterior mode of implies that
there is a roughly 0.3% chance of an exogenous
break-up in negotiations when a wage offer is rejected. Third, the
posterior modes of our model parameters, along with the assumption
that the steady state unemployment rate equals 5.5%, implies that it costs firms 0.27 days
of marginal revenue to prepare a counteroffer during wage
negotiations (see Table 5). Fourth, the posterior mode of steady
state hiring costs as a percent of total wages of newly hired
workers is equal to 6.7%.13 Silva and Toledo
(2009) report that, depending on the exact costs included, the
value of this statistic is between 4 and
14 percent, a range that encompasses the
corresponding statistic in our model. Fifth, the posterior mode for
the replacement ratio is 0.67. Based on a summary of
the literature, Gertler, Sala and Trigari (2008) argue that a
plausible range for the replacement ratio is 0.4 to 0.7. The lower bound is based on studies
of unemployment insurance benefits, while the upper bound takes
into account informal sources of insurance.14 Sixth, the
posterior mode of
governing the
responsiveness of
to
technology shocks, is close to zero (0.014). So,
these variables are very unresponsive in the short-run to
technology shocks. Interestingly, the posterior mode for
is relatively large compared to
the posterior of the other
parameters If we set
to 0.014, then
unemployment benefits initially move by very little after a
technology shock. Other things equal, the model then generates a
rise in employment and falls in unemployment and inflation that are
too large relative to the responses emerging from the VAR. This
result is one way to see that our model has no difficulty in
resolving the Shimer puzzle. We return to this important point
below. Seventh, the posterior modes of the parameters governing
monetary policy are similar to those reported in the literature
(see for example Justiniano, Primiceri, and Tambalotti, 2010).

The solid black lines in Figures 3-5 present the impulse
response functions to monetary policy shock, a neutral-technology
shock and an investment-specific technology shock, implied by the
estimated VAR. The grey areas represent 95 percent probability
intervals. The solid lines with the circles correspond to the
impulse response functions of our model evaluated at the posterior
mode of the structural parameters. Figure 3 shows that the model
does very well at reproducing the estimated effects of an
expansionary monetary policy shock, including the hump-shaped rise
of real GDP and hours worked and the muted response of inflation.
Notice that real wages respond by much less than hours to the
monetary policy shock. Even though the maximal rise in hours worked
is roughly 0.14%, the maximal rise in real wages
is only 0.06%. Significantly, the model accounts
for the hump-shaped fall in the unemployment rate as well as the
rise in the job finding rate and vacancies that occur after an
expansionary monetary policy shock. The model does understate the
rise in the capacity utilization rate. The sharp rise of capacity
utilization in the estimated VAR may reflect that our data on the
capacity utilization rate pertains to the manufacturing sector,
which may overstate the average response across all sectors in the
economy.

From Figure 4 we see that the model does a good job of
accounting for the estimated effects of a neutral technology shock.
Of particular note is that the model reproduces the estimated sharp
fall in the inflation rate that occurs after a positive neutral
technology shock, a feature of the data stressed in Altig,
Christiano, Eichenbaum and Linde (2011) and Paciello (2009). Also,
the model generates a sharp fall in the unemployment rate along
with a large rise in job vacancies and the job finding rate.
Finally, Figure 5 shows that the model does a good job of
accounting for the estimated response of the economy to an
investment-specific technology shock.

6.2. The Estimated Sticky Wage
Model

In this subsection we discuss the empirical properties of the
sticky wage model and compare its performance to the 'Alternating
Offer, Hiring' model. Recall that our sticky wage model rules out
indexation of wages to technology and inflation. We comment on a
version of the model that allows for such indexation at the end of
the subsection.

Table 3 reports parameter values of the 'Sticky Wage' model that
we set a priori. Note in particular that we fix to 0.75 so that wages change on average
once a year.15 Table 4 reports the posterior modes
of the estimated sticky wage model parameters. Several results
emerge from the table. First, the posterior mode of the coefficient
on inflation in the Taylor rule, is
substantially higher than the corresponding posterior mode in the
'Alternating Offer, Hiring' model (2.09 versus
1.36). Second, the degree of price
stickiness is higher in the 'Sticky Wage' model than in the
'Alternating Offer, Hiring' model. In the sticky wage model, prices
are estimated to change on average roughly once a year compared to
2.4 quarters in the'Alternating Offer,
Hiring' model .

Figures 3-5 show that with two important exceptions, the sticky
wage model does reasonably well at accounting for the estimated
impulse response functions. These exceptions are that the model
understates the responses of inflation to a neutral technology
shock and a monetary policy shock.

We want to compare the fit of our baseline model with that of
the sticky wage model. The marginal likelihood is a standard
measure of fit. However, using it here is complicated by the fact
that the two models do not address the same data. For example, the
sticky wage model has no implications for vacancies and the job
finding rate.16 To obtain a measure of fit based on
a common data set, we integrate out unemployment, the job finding
rate and vacancies from the marginal likelihood associated with our
baseline model.17 The marginal likelihoods based on
the impulse response functions of the nine remaining variables are
reported in Table 4 (see 'Laplace, 9 Variables'). The marginal
likelihood for our baseline model is 67 log points higher than it
is for the sticky wage model. We conclude that, subject to the
approximations that we used to compute the marginal likelihood
function, there is substantial statistical evidence in favor of the
'Alternating Offer, Hiring' model relative to the sticky wage
model.

Finally, we estimated a version of the sticky wage model where
we allow for wage indexation. In particular, we assume that if a
labor supplier cannot re-optimize his wage, it changes by the
steady state growth rate of output, times the
lagged inflation rate. Table 4 reports that, relative to the no
indexation sticky wage model, the posterior modes of many of the
model parameters move towards those reported for the 'Alternating
Offer, Hiring' model. See for example the habit persistence
parameter the Taylor rule parameter and the markup parameter The
impulse response functions of our model and the sticky wage model
with indexation are qualitatively very similar, Table 4 indicates
that the marginal likelihood of the latter model is about 4 log
points lower than our baseline model. Overall, we conclude that the
performance of the two models is similar. But the performance of
the sticky wage model depends very much on the troubling wage
indexation assumption.

6.3. The Estimated DMP Model

In this subsection, we compare the performance of our version of
the DMP model with the 'Alternating Offer, Hiring' model. Recall
that there are two differences between these models: the assumption
of hiring versus search costs and the way that wages are
determined. To assess the importance of each difference, we proceed
as follows. First, we modify the baseline model by replacing the
alternating offer bargaining specification with the Nash sharing
rule of the DMP model (see subsection 4.4.1). We
consider two cases: one with search costs (the 'Nash Sharing,
Search' model) and one with hiring costs (the 'Nash Sharing,
Hiring' model). Second, we compare the empirical performance of
these models. Finally, we isolate the role of hiring versus search
costs in the 'Alternating Offer, Hiring' model by considering a
version of this model in which hiring costs are replaced by search
costs, the 'Alternating Offer, Search' model. This version of the
model is closest in spirit to HM who assume that there are search
costs rather than hiring costs.

The solid black lines in Figures 6-8 are the impulse responses,
implied by the estimated VAR, to a monetary policy shock, a neutral
technology shock and an investment-specific technology shock. The
grey areas represent 95 percent probability intervals. The solid
lines with the circles, the dashed lines, the dashed lines broken
by dots and the thick solid line correspond to the impulse response
functions of the 'Alternating Offer, Hiring', 'Nash Sharing,
Hiring', 'Nash Sharing, Search', and 'Alternating Offer, Search'
models, respectively. All impulse response functions are
constructed using the posterior mode of the structural parameters
estimated for the 'Alternating Offer, Hiring' model. Conditional on
the values of the other structural parameters, we calibrate the
value of in the Nash models to obtain a steady
state rate of unemployment equal to 5.5%. The
values of in the 'Nash Sharing, Search' and
'Nash Sharing, Hiring' models are 0.48 and
0.66, respectively.

From Figure 6 we see that the responses of output, hours worked,
job finding rates, unemployment, vacancies, consumption and
investment to a monetary policy shock are weakest in the 'Nash
Sharing, Search' model. That model also gives rise to the strongest
real wage and inflation responses. These findings are closely
related to the Shimer (2005a) critique of the DMP model.

Comparing the 'Nash Sharing, Search' and 'Nash Sharing, Hiring'
models we see that switching from the search cost to the hiring
cost specification improves the performance of the model. In
particular, output, job finding rates, unemployment, vacancies,
consumption and investment exhibit stronger responses to a monetary
policy shock, while real wages and inflation exhibit weaker
responses. The basic intuition is that the search cost
specification implies that yields on posting vacancies are
countercyclical. This force mutes the effects of an expansionary
monetary policy shock. A similar result emerges comparing the
'Alternating Offer, Hiring' and 'Alternating Offer, Search'
models.

From Figure 7 we see that the weakest responses of output, hours
worked, job finding rates, unemployment, vacancies, consumption and
investment to a neutral technology shock arise in the 'Nash
Sharing, Search' model. Also, consistent with Shimer (2005a),
vacancies, job finding rates and unemployment are essentially
unresponsive to the shock. As in Figure 6, moving from a search
cost to a hiring cost specification improves the performance of the
model. Finally, Figure 8 shows that similar but less dramatic
conclusions emerge from considering an investment-specific
technology shock.

We now consider the results of estimating the 'Nash Sharing,
Search' and 'Nash Sharing, Hiring' models. Consider first the
posterior mode of the estimated structural parameters (see Table
4). The key result here is that for the 'Nash Sharing, Search' and
the 'Nash Sharing, Hiring' models, the posterior modes of the
replacement ratio are 0.96 and 0.90, respectively. The high value of the replacement ratio
enables the Nash sharing models to account for the response of
unemployment to the three structural shocks that we consider.
Indeed the impulse response functions of the 'Alternating Offer,
Hiring' and the two Nash Sharing models are visually similar (see
Figures A1 - A3 in the technical appendix). This finding is
reminiscent of Hagedorn and Manovskii's (2008) argument that a high
replacement ratio has the potential to boost the volatility of
unemployment and vacancies in search and matching models.

The 'Alternating Offer, Hiring' model outperforms all Nash
models, based on the marginal likelihood. Table 4 reports that the
marginal likelihood for that model is 38 and
27 log points higher than it is for the
'Nash Sharing, Search' and 'Nash Sharing, Hiring' models,
respectively.18 We infer that, subject to the
approximations that we have made in calculating the marginal
likelihood function, there is substantial statistical evidence in
favor of the 'Alternating Offer, Hiring' model.

Finally, we investigate the relative importance of hiring versus
search costs in our preferred model. To this end, we estimated the
'Alternating Offer, Search' model. From Table 4, we see that there
are two significant changes in the posterior mode of the structural
parameters relative to those of the 'Alternating Offer, Hiring'
model. First, the probability of a bargaining-breakup falls from
0.3% percent to 0.06%. So,
the outside option, has a more limited
effect on bargaining. Second, search costs are driven to a
relatively low value as a percent of gross output, 0.17%. The latter value corresponds to a share of search
costs relative to the wages of new hires wages of about 2%. This value lies below the lower bound of the range
suggested in Silva and Toledo (2009). In effect, the search part of
the 'Alternating Offer, Search' model is driven out of the model.
From Table 4 we see that the marginal likelihood for our baseline
model is 9 log points higher than it is for the
'Alternating Offer, Search' model. These results imply that moving
from search to hiring costs improves the empirical performance of
the model. An additional reason to favor the hiring cost
specification comes from the micro evidence in Yashiv (2000),
Cheremukhin and Restrepo-Echavarria (2010) and Carlsson, Eriksson
and Gottfries (2013). Third, and most importantly, the improvement
from moving from 'Nash Sharing' models to 'Alternating Offer'
models is larger than the improvement due to moving from search to
hiring costs in the 'Alternating Offer' models (see for example the
marginal likelihood values in Table 4).

6.4. The Cyclical Behavior of Unemployment
and Vacancies

We have argued that our model can account for the estimated
response of unemployment and vacancies to monetary policy shocks,
neutral technology shocks and investment-specific technology
shocks. Our methodology is quite different than the one used in
much of the relevant labor market search literature. In this
subsection we show that our model also does well when we assess its
performance using the procedures adopted in that literature. Shimer
(2005a) considers a real version of the standard DMP model in which
labor productivity shocks and the job separation rate are exogenous
stationary stochastic processes. He argues that shocks to the job
separation rate cannot be very important because they lead to a
positively sloped Beveridge curve.

Shimer (2005a) deduces the model's implications for HP-filtered
moments which he compares to the analog moments in U.S. data. He
focuses on the volatility of vacancies divided by unemployment
relative to the volatility of labor productivity. Shimer also looks
at the persistence and the correlation among these
variables.19 Shimer (2005a) emphasizes that the
standard DMP model fails to account of the volatility of vacancies
divided by unemployment,
relative to the volatility of
labor productivity,
. We refer to the ratio
as the 'volatility
ratio'.20 Shimer (2005a) reports that the
volatility ratio is about 20 in U.S. data. But
in the standard DMP model analyzed by Shimer (2005a), the
volatility ratio is only roughly 2.

In the spirit of Shimer's (2005a) analysis, we consider a
version of our model in which the only source of uncertainty is a
neutral technology shock, . As in the
estimated DSGE model we assume that the growth rate of the neutral
technology shock is . We set the standard
deviation of the innovation to that shock,
to 0.7%, a
value that implies that the standard deviation of HP-filtered real
GDP in the model and the data are the same.21

Table 6 reports estimates for various moments of the data and
the implications of the 'Alternating Offer, Hiring' and 'Nash
Sharing, Search' models for these moments. Consider first the
results of calculating model moments using parameter values equal
to the estimated posterior mode of the 'Alternating Offer, Hiring'
model. The key finding is that the volatility ratio implied by the
'Alternating Offer, Hiring' model is 27.8 which
effectively reproduces the analog statistic in our data, i.e.
27.6. In this sense our model is not
subject to the Shimer critique. Notice that our model also accounts
very well for the standard deviations and, with one exception, the
first-order autocorrelations of vacancies, unemployment and labor
productivity. The exception is that the model somewhat understates
the first-order autocorrelation of vacancies. Finally, the model
does quite well in reproducing the unconditional correlations
between vacancies, unemployment and labor productivity. Consider
next the implications of the 'Nash Sharing, Search' model.
Consistent with Shimer (2005a), this model generates a much smaller
value of the volatility ratio, namely 13.2. In
addition, the model has problems reproducing volatilities, as well
as some of the first-order autocorrelations and correlation
statistics. For example, the model generates the wrong sign for the
correlation between unemployment and labor productivity
(0.2 in the model and -0.3 in
the data).

Interestingly, the volatility ratio implied by the 'Nash
Sharing, Search' is higher than the one reported in Shimer (2005a)
for the DMP model. The difference in results reflects that our
medium-sized DSGE model is considerably more complex than the model
used by Shimer (2005a). We have examined the case when we eliminate
habit formation, the working capital channel and physical capital
from our model. Further, we also suppose that firms change prices
roughly once a quarter (
Under these assumptions - which
brings our model as close as possible to the one studied by Shimer
(2005a) - it turns out that the 'volatility ratio' is equal to
24.4 in the 'Alternating Offer, Hiring' and
only 2.6 in the 'Nash Sharing, Search' model.

Finally, we evaluate the implications of the 'Nash Sharing,
Search' model using the posterior mode of the parameter estimates
for that model. Among other things, the replacement ratio for this
model is 0.96. Consistent with Hagedorn and Manovskii
(2008), we find that the 'Nash Sharing, Search' model is able to
roughly account for the 'volatility ratio'. That ratio is
27.6 in our data and 24.5 in
the 'Nash Sharing, Search' model.

Hagedorn and Manovksii (2008) and Shimer (2005a) work with
stationary representations of the neutral technology shock. To
assess the robustness of our results we redid our calculations
assuming that is an AR(2) process with roots
equal to 1.35 and -0.45 and a standard
deviation of the shock equal to 0.3. With these
parameter values the estimated 'Alternating Offer, Hiring' model
roughly matches the point estimates of the four volatility moments,
the four first-order autocorrelation moments and the six
correlation moments reported in Table 6. Importantly, the
'volatility ratio' in the model is roughly equal to 30, while it is 27.6 in the data. In
contrast, with this parameterization of the technology process, the
'volatility ratio' in the 'Nash Sharing, Search' model is only
equal to 10. If we work with the mode of the
estimated version of the latter model, we are able to account for
the 'volatility ratio'. So, the results that we obtain with the
stationary technology process are very similar to those reported
above.

Viewed as a whole, the results of this section corroborate our
argument that the 'Alternating Offer, Hiring' model does well at
accounting for the cyclical properties of key labor market
variables and outperforms the competing models that we consider.
The result holds whether we assess the model using our impulse
response methodology or use the statistics stressed in the relevant
literature.

7. Conclusion

This paper constructs and estimates an equilibrium business
cycle model which can account for the response of the U.S. economy
to neutral and investment-specific technology shocks as well as
monetary policy shocks. The focus of our analysis is how labor
markets respond to these shocks. Significantly, our model does not
assume that wages are sticky. Instead, we derive inertial wages
from our specification of how firms and workers interact when
negotiating wages. This inertia can be interpreted as applying to
the period-by-period wage, or to the present value of the wage
package negotiated at the time that a worker and firm first meet.
It remains an open question which implications for optimal policy
of existing DSGE models are sensitive to abandoning the sticky wage
assumption. We leave the answer to this question to future
research.

We have been critical of standard sticky wage models in this
paper. Still, Hall (2005) describes one interesting line of defense
for sticky wages. He introduces sticky wages into the DMP framework
in a way that satisfies the condition that no worker-employer pair
has an unexploited opportunity for mutual improvement (Hall, 2005,
p. 50). A sketch of Hall's logic is as follows: in a model with
labor market frictions, there is a gap between the reservation wage
required by a worker to accept employment and the highest wage a
firm is willing to pay an employee. This gap, or bargaining set,
fluctuates with the shocks that affect the surplus enjoyed by the
worker and the employer. When calibrated based on aggregate data,
the fluctuations in the bargaining set are sufficiently small and
the width of the set is sufficiently wide, that an exogenously
sticky wage rate can remain inside the set for an extended period
of time. Gertler and Trigari (2009) and Shimer (2012b) pursue this
idea in a calibrated model while Gertler, Sala and Trigari (2008)
do so in an estimated, medium-sized DSGE model.22 A concern about
this strategy for justifying sticky wages is that the microeconomic
shocks which move actual firms' bargaining sets are far more
volatile than what the aggregate data suggest. As a result, it may
be harder to use the preceding approach to rationalize sticky wages
than had initially been recognized. An important task is to
discriminate between the approach taken in this paper and the
approach proposed in Hall (2005).

We wish to emphasize that our approach follows HM in assuming
that the cost of disagreement in wage negotiations is relatively
insensitive to the state of the business cycle. This assumption
played a key role in the empirical success of our model. Assessing
the empirical plausibility of this assumption using microeconomic
data is a task that we leave to future research.

References

Aguiar, Mark, Erik Hurst and Loukas Karabarbounis, 2012, "Time Use During the Great Recession," American Economic
Review, forthcoming.

Christiano, Lawrence J., Martin Eichenbaum, and Charles L.
Evans, 1999, "Monetary Policy Shocks: What Have We Learned and to
What End?," Handbook of Macroeconomics, edited by John B.
Taylor and Michael Woodford, pages 65-148, Amsterdam, New York and
Oxford: Elsevier Science, North-Holland.

Appendix

A. The Equilibrium Wage Rate

We develop an analytic expression relating the equilibrium wage
rate to economy-wide variables taken as given by firms and workers
when bargaining.

It is useful to re-state the indifference conditions for the
worker and the firm given in the main text:

for

for

for

Rewrite the previous expressions and abbreviate variables taken
as given during the wage bargaining:

for

for

for

Or, in short:

for

for

Write out:

Substituting several times results in the following pattern:

Rearrange:

Or, equivalently:

(A.1)

Note that:

Substract and rearrange:

So that:

(A.2)

Using (A.2), we can write the square brackets
multiplying and in (A.1) as:

and

Hence,

(A.3)

The square bracket in the last line in (A.3)
can be written as,

Note that differentiating both sides of:

yields

Hence, the square bracket of the last line in (A.3) can be expressed more compactly as:

Finally, the terms involving in (A.3) can be rewritten as:

Pulling everything together, we can write (A.3) as:

Collecting terms gives:

Simplifying, using straightforward algebra yields:

After some further rewriting, we can express the previous
expression as the following alternating offer bargaining sharing
rule:

where

Note that
Alternatively, we can write the alternating offer bargaining
sharing rule in terms of the following variables:

Finally, notice that for
the sharing rule
becomes:

B. Marginal Likelihood for a Subset of
Data

We derive the marginal likelihood function for a common subset
of data. We denote the full set of data by the vector,
We decompose
into two parts:

where
is
and
We have a marginal likelihood
for

where
denotes the likelihood of
conditional on the model
parameters, Also,
denotes the priors.
We seek the marginal likelihood for the subset of data
which is defined as:

For this, we rely heavily on the Laplace approximation to

(B.1)

where
denotes the
second derivative of
with respect to evaluated at the mode,
Also, denotes
the number of elements in Note that we
can write the matrix as follows:

Where is the upper
block of
and is the lower
block. The zeros on the
off-diagonal of reflect our assumption that
is diagonal. Using this notation, we write
our approximation to the likelihood (5.3)
as follows:

Substituting this expression into (B.1), we
obtain the following representation of the marginal likelihood of

Now it is straightforward to compute our approximation to

Here, we have used

which follows from the fact that the integrand is a density
function.

Table 1a: Parameters and Steady State Values in the Small Macro Model - Panel A: Parameters

Parameter

Value

Description

ß

1.03-0.25

Discount factor

ξ

2/3

Calvo price stickiness

λ

1.2

Price markup parameter

ΡR

0.7

Taylor rule: interest rate smoothing

rπ

1.7

Taylor rule: inflation coefficient

ry

0.1

Taylor rule: employment coefficient

Ρ

0.9

Job survival probability

M

60

Max. bargaining rounds per quarter

δ

0.005

Probability of bargaining session break-up

Γ

0.95

AR(1) technology

a Sticky wages as in Erceg, Henderson and Levin (2000).

Table 1b: Parameters and Steady State Values in the Small Macro Model - Panel B: Steady State Values

Table 4: Priors and Posteriors of Parameters for the Medium-Sized Model

Model #

Prior (D,Mean,Std)

Prior: Alternating Offer Bargaining: HiringΜ1(Mode,Std)

Prior: Alternating Offer Bargaining: Search Μ2(Mode,Std)

Posterior: Nash Sharing: HiringΜ3(Mode,Std)

Posterior: Nash Sharing: SearchΜ4(Mode,Std)

Posterior: Wage Indexation: No Μ5(Mode,Std)

Posterior: Wage Indexation: YesΜ6(Mode,Std)

Price Setting Parameters: Price Stickiness ξ

B,0.66,0.15

0.58,0.03

0.64,0.04

0.70,0.02

0.74,0.02

0.74,0.02

0.65,0.03

Price Setting Parameters: Price Markup Parameter λ

G,1.20,0.05

1.43,0.04

1.43,0.04

1.42,0.04

1.43,0.04

1.25,0.05

1.36,0.04

Monetary Authority Parameters: Taylor Rule: Smoothing ρR

B,0.70,0.15

0.86,0.01

0.86,0.01

0.86,0.01

0.84,0.01

0.78,0.01

0.86,0.01

Monetary Authority Parameters: Taylor Rule: Inflationrπ

G,1.70,0.15

1.36,0.11

1.36,0.11

1.39,0.12

1.37,0.12

2.09,0.15

1.48,0.13

Monetary Authority Parameters: Taylor Rule: GDPry

G,0.10,0.05

0.04,0.01

0.04,0.01

0.04,0.01

0.04,0.01

0.01,0.01

0.09,0.03

Preferences and Technology: Consumption Habitb

B,0.50,0.15

0.83,0.01

0.83,0.01

0.82,0.01

0.82,0.01

0.70,0.02

0.76,0.02

Preferences and Technology: Capacity Util. Adj. Costσa

G,0.50,0.30

0.08,0.04

0.06,0.03

0.06,0.04

0.05,0.03

0.04,0.02

0.04,0.03

Preferences and Technology: Investment Adj. CostS"

G,8.00,2.00

13.67,1.8

13.80,1.9

13.41,1.9

13.50,1.9

5.31,0.82

7.94,1.10

Preferences and Technology: Capital Shareα

B,0.333,0.03

0.24,0.02

0.24,0.02

0.25,0.02

0.25,0.02

0.32,0.02

0.29,0.02

Preferences and Technology: Techn. Diffusion g, φ, κ, γθi

B,0.50,0.20

0.01,0.01

0.02,0.01

0.01,0.01

0.01,0.01

0.04,0.02

0.02,0.01

Preferences and Technology: Technology Diffusion D θD

B,0.50,0.20

0.74,0.15

0.66,0.19

0.12,0.03

0.10,0.02

-

-

Labor Market Parameters: Prob. of Barg. Breakup100δ

G,0.50,0.40

0.30,0.06

0.06,0.06

-

-

-

-

Labor Market Parameters: Replacement RatioD/w

B,0.40,0.10

0.67,0.06

0.69,0.07

0.90,0.01

0.96,0.01

-

-

Labor Market Parameters: Hiring-Search Cost/Ysl

G,1.00,0.30

0.50,0.16

0.17,0.04

0.55,0.17

0.47,0.14

-

-

Labor Market Parameters: Match. Function Param. σ

B,0.50,0.10

0.56,0.03

0.52,0.04

0.56,0.03

0.50,0.04

-

-

Labor Market Parameters: Inv. Labor Supply Elast. ψ

G,1.00,0.25

-

-

-

-

0.89,0.20

2.19,0.33

Shocks: Std. Monetary Policy σR

G,0.65,0.05

0.60,0.03

0.62,0.03

0.62,0.03

0.62,0.03

0.64,0.04

0.62,0.03

Shocks: Std. Neutral Technology σμz

G,0.10,0.05

0.14,0.01

0.14,0.02

0.17,0.01

0.17,0.02

0.31,0.02

0.25,0.02

Shocks: Std. Invest. Technology σψ

G,0.10,0.05

0.11,0.02

0.11,0.02

0.11,0.02

0.11,0.02

0.15,0.02

0.14,0.02

Shocks: AR(1) Invest. Technology ρψ

B,0.75,0.10

0.73,0.06

0.73,0.06

0.74,0.06

0.75,0.05

0.58,0.06

0.63,0.06

Memo Items: Log Marg. Likelihood (Laplace, 12 Variables):

302.0

291.5

297.3

263.5

-

-

Memo Items: Log Marg. Likelihood (Laplace, 9 Variablesb):

328.2

319.1

301.1

290.3

260.9

324.5

Memo Items: Post. Odds - Μ1 : Μi,i = 1, .., 6 (9 Variab.):

1:1

9e3:1

6e11:1

3e16:1

2e29:1

1:40.4

Notes: sl denotes the steady state hiring or search cost to gross output ratio (in percent): For model specifications where particular
parameter values are not relevant, the entries in this table are blank.
α Sticky wage model as in Erceg, Henderson and Levin (2000).b Common dataset across all models, i.e. when unemployment, vacancies and job finding rates are excluded.

Notes: u, v and Y/l denote the unemployment rate, vacancies and labor productivity; is the standard
deviation of these variables. All data are in log levels and hp-filtered with smoothing parameter 1600. The
sample period is 1951Q1 to 2008Q4. Data sources are the same as those used for the estimation of the
medium-sized model. Similar to Shimer (2005), we simulate the model using a unit-root neutral technology
shock: See the main text for details.

Notes: u, v and Y/l denote the unemployment rate, vacancies and labor productivity; is the standard
deviation of these variables. All data are in log levels and hp-filtered with smoothing parameter 1600. The
sample period is 1951Q1 to 2008Q4. Data sources are the same as those used for the estimation of the
medium-sized model. Similar to Shimer (2005), we simulate the model using a unit-root neutral technology
shock: See the main text for details.

Notes: u, v and Y/l denote the unemployment rate, vacancies and labor productivity; is the standard
deviation of these variables. All data are in log levels and hp-filtered with smoothing parameter 1600. The
sample period is 1951Q1 to 2008Q4. Data sources are the same as those used for the estimation of the
medium-sized model. Similar to Shimer (2005), we simulate the model using a unit-root neutral technology
shock: See the main text for details.

Footnotes

**
The views expressed in this paper are those of the authors and do
not necessarily reflect those of the Board of Governors of the
Federal Reserve System or of any other person associated with the
Federal Reserve System. Return to
text

1. For discussions of high labor supply
elasticities in real business cycle models, see for example,
Rogerson and Wallenius (2009) and Chetty, Guren, Manoli and Weber
(2012). For discussions of the role of high replacement ratios in
DMP models see for example, Hagedorn and Manovskii (2008) and
Hornstein, Krusell and Violante (2010). Return to text

5. For a paper that uses a reduced form
version of HM in a calibrated real business cycle model, see
Hertweck (2006). Return to text

6. This perspective on bargaining has been
stressed in Rubinstein (1982), Binmore (1985) and Binmore,
Rubinstein and Wolinsky (1986). Return
to text

7. A technical appendix is available at:
https://sites.google.com/site/mathiastrabandt/home/downloads/CETtechapp.pdf
sites.google.com/site/mathiastrabandt/home/downloads/CETtechapp.pdf
. Return to text

8. For a different view, see Fujita and
Ramey (2009) who argue that the job separation rate is
countercyclical. Return to text

9. Note that in this expression (and
elsewhere) the firm's outside option does not have to take into
account that it is costly for the firm to meet another worker at
the start of period This reflects that in
our environment the firm's decision to undertake an expense to meet
a worker is unrelated to its labor market experience in previous
periods. Return to text

10. Denote the probability that a worker
separates from a job at a monthly rate by
The probability that a
person employed at the end of a quarter separates in the next three
months is
. Shimer (2012a) reports that
implying a quarterly
separation rate of 0.0986. HM assume a similar value of 0.03 for
the monthly separation rate. This value is also consistent with
Walsh's (2003) summary of the empirical literature. Return to text

11. See CTW for a sensitivity analysis
with respect to the lag length of the VAR. Return to text

12. See section A of the technical
appendix in CTW for details about the data. Return to text

13. Table 4 reports the hiring cost to
gross output ratio in steady state which is defined as:
. Here
is equal to
evaluated at
steady state. Given and the real wage,
it is straightforward to compute hiring
costs as a share of the wage of newly hired workers:
Return to text

14. Aguiar, Hurst and Karabarbounis
(2012) find that workers who become unemployed increase the amount
of time that they spend on home production by roughly 30 percent. This increase in home production could
potentially rationalize a replacement ratio that is higher than
0.7. Return to
text

15. We encountered numerical problems in
calculating the posterior mode of model parameters when we did not
place a dogmatic prior on . We suspect
that this problem stems from indeterminacy of the equilibrium for
various configurations of the parameter values. As Ascari, Benzoni
and Castelnuovo (2011) stress, the range of parameter values for
which the indeterminacy problem arises is substantially larger in
sticky wage models without indexation relative to models with
indexation. Return to text

16. Galí (2011) has shown how to
derive implications for the unemployment rate from the sticky wage
model. For a discussion of this approach see Gal í, Smets
and Wouters (2012), Christiano (2012) and Christiano, Trabandt and
Walentin (2012). Return to text

17. See Appendix B for a detailed
derivation of how we integrate out these variables. Return to text

18. Using different models estimated on
macro data of various countries, Christiano, Trabandt and Walentin
(2011b), Furlanetto and Groshenny (2012a,b) and Justiniano and
Michelacci (2011) also conclude that a hiring cost specification is
preferred to a search cost specification. Return to text

20. Here,
denotes the
standard deviation of a time series variable after it has been
HP-filtered. Return to text

21. This value of
is much larger than the
mode of the posterior in the estimated medium-sized DSGE model.
This fact reflects that in this subsection we are attributing all
movements in real GDP to neutral technology shocks. We do not
maintain such an assumption when we estimate the model. With such a
large value of
there are occasions in a
long simulated time series where variables like the unemployment
rate rise to high levels not observed in the data. Return to text

22. See also Krause and Lubik (2007),
Christiano, Ilut, Motto and Rostagno (2008) and Christiano,
Trabandt and Walentin (2011b). Return to
text